 |
Center for Nonlinear Studies |
We show how growth processes of Laplacian type can be simulated by statistical mechanics of 2D Coulomb charges in an external field (a logarithmic gas). The growing cluster is represented by a domain where the mean density of the charges does not vanish in the large N limit, and the physical growth time is identified with a coupling constant of the external field. The logarithmic gas picture applies both to Laplacian growth of smooth domains in the plane and to the growth of slit domains described by the Loewner equation. It also provides a key to integrable structure of the models and gives a unique way of their discretization or 'quantization' preserving integrability. From this point of view, we discuss growth models associated to the Toda lattice and KP integrable hierarchies.